Escape the Klotski dungeon!

Question

Help! I am trapped in a dungeon with multiple rooms full of sliding tiles! I need to find a way of sliding the tiles such that the 2x2 tile can be freed, by sliding it out of the exit on the North side of the dungeon.
There are some restrictions: every tile in a given room can only be slid if the 2x2 tile is completely inside that room, and that is the only tile that can traverse a passageway: ==.

Here is the floor-plan of the dungeon:

                  │ ░░░░░░░ │
              ┌───┘·········└───┐
              │     ┌─────┐     │
              │     └─────┘     │   119
              │ ┌─────┐ ┌─────┐ │
              │ └─────┘ └─────┘ │
              │ ┌─┐ ┌─┐ ┌─┐ ┌─┐ │
              │ │ │ └─┘ └─┘ │ │ │
              │ │ │         │ │ │
              │ └─┘         └─┘ │
              │ ┌─┐         ┌─┐ │
              │ └─┘         └─┘ │
              └───┐·········┌───┘
                  │ ░░░░░░░ │
              ┌───┘·········└───┐
              │ ┌─────┐ ┌─┐ ┌─┐ │
              │ └─────┘ │ │ │ │ │    54
              │ ┌─────┐ │ │ │ │ │
              │ └─────┘ └─┘ └─┘ │
              │ ┌─────┐ ┌─┐ ┌─┐ │
              │ └─────┘ └─┘ └─┘ │
              │ ┌─┐             │
              │ └─┘             │
              │ ┌─┐             │
              │ └─┘             │
              └───┐·········┌───┘
                  │ ░░░░░░░ │
              ┌───┘·········└───┐
              │ ┌─┐ ┌─┐ ┌─────┐ │
              │ │ │ │ │ └─────┘ │    41
              │ │ │ │ │ ┌─┐ ┌─┐ │
              │ └─┘ └─┘ └─┘ └─┘ │
              │ ┌─────┐ ┌─────┐ │
              │ └─────┘ └─────┘ │
              │ ┌─┐ ╔═════╗     │
              │ └─┘ ║     ║     │
              │ ┌─┐ ║     ║     │
              │ └─┘ ╚═════╝     │
              └─────────────────┘

Note: The 2x2 tile which must be freed is represented with a double outline. The number next to each room can be considered as a difficulty level, as it is the required number of single piece single unit moves.

Here is a more compact version, useful for shortening answers:

 |==|
++==++
| HH |
|HHHH|
|VSSV| 119
|V  V|
|S  S|
++==++
 |==|
++==++
|HHVV|
|HHVV|
|HHSS| 54
|S   |
|S   |
++==++
 |==|
++==++
|VVHH|
|VVSS|
|HHHH| 41
|SRR |
|SRR |
+----+

Answer 1

Bottom Puzzle:

Middle Puzzle:

Top Puzzle: _{^{(AKA: Pain and a half)}}

Due to the sheer number of steps, I had to animate this one.
It loops forever so, if you missed the beginning, just wait.
 

 
My solution takes 119 steps.
I saved 10 steps thanks to Zizy Archer but OP has pointed out where I actually skipped steps in several places. I'm doing this all in Excel and never programmatically checked it. I added a helper to my original file and think I ferreted out all the issues. Every move either changes the color of two squares (single block moving or long block moving along its long axis) or two squares (long block moving along its short axis or big block moving). I programmatically found and then manually checked all the four square color changes. However, as OP has proven so excellent at finding my errors, here's a slower version of the GIF above in case someone else can find an error. That seems likely since I'm still two moves shorter than OP's solver that did the entire thing programmatically.

_{The steps count a two square movement in the same direction as two steps and does not include the first and last frames (starting position and escaped position). Per OP, this is how their solver counted as well.}